## Paper: Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds (at LICS 2004)

**Sanjit A. Seshia Randal E. Bryant**

### Abstract

Given a formula Φ in quantifier-free Presburger arithmetic, it is well known that, if there is a satisfying solution to Φ, there is one whose size, measured in bits, is polynomially bounded in the size of Φ. In this paper, we consider a special class of quantifier-free Presburger formulas in which most linear constraints are separation (difference-bound) constraints, and the non-separation constraints are sparse. This class has been observed to commonly occur in software verification problems. We derive a new solution bound in terms of parameters characterizing the sparseness of linear constraints and the number of non-separation constraints, in addition to traditional measures of formula size. In particular, the number of bits needed per integer variable is linear in the number of non-separation constraints and logarithmic in the number and size of non-zero coefficients in them, but is otherwise independent of the total number of linear constraints in the formula. The derived bound can be used in a decision procedure based on instantiating integer variables over a finite domain and translating the input quantifier-free Presburger formula to an equi-satisfiable Boolean formula, which is then checked using a Boolean satisfiability solver. We present empirical evidence indicating that this method can greatly outperform other decision procedures.

### BibTeX

@InProceedings{SeshiaBryant-DecidingQuantifierF, author = {Sanjit A. Seshia and Randal E. Bryant}, title = {Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds}, booktitle = {Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science (LICS 2004)}, year = {2004}, month = {July}, pages = {100--109}, location = {Turku, Finland}, publisher = {IEEE Computer Society Press} }